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Math 176 Calculus – Sec. 5.6: The Logarithm Defined as an Integral
I. Review
A. Relationship between Logarithms and Exponents
y = b x ⇔ x = log b y
B. Properties of Logarithms
For any numbers b, m, n >0
1. log b (mn) = log b (m) + log b (n)
⎛m⎞
2. log b ⎜ ⎟ = log b (m) − log b (n)
⎝n⎠
( )
3. log b m r = r log b (m)
⎛1⎞
4. log b ⎜ ⎟ = − log b (m)
⎝m⎠
5. log b (b ) = 1
6. log b (1) = 0
C. In words: The log b y = x : x is the power to which the base (b) must be raised to get
a given number (y).
D. Common Notation
1. log 10 ( x) = log( x )
2. log e ( x ) = ln( x )
Log base e is the “natural logarithm” function.
II The natural logarithm function.
A. A definition of the natural logarithm function:
x
1
ln( x) = ∫ dt ,
t
1
x>0
1
and y = ln( x ) , x > 0
x
1
1. Graph of y =
, x >0
x
B. Graphs of
y=
2. Graph of
y = ln( x ) , x > 0
C. Interpretation of ln(x) :
1. If x > 1, then ln(x) is the area under the curve of
y=
1
from t=1 to t=x.
t
1
2.
1
ln (1) = ∫ dt = 0 since the upper and lower limits are the same.
t
1
3. If 0 < x < 1, then ln(x) gives the negative of the area under the curve from t=x to t=1.
x
4. ln(x) is not defined for x<0.
5. The domain of ln(x) is
6.
(
)
(0,∞ ) and the range is (−∞,∞ ) .
(
)
lim ln ( x ) → ∞ , lim+ ln( x ) → - ∞
x→∞
x →0
7. The function is continuous, increasing, and one-to-one.
8. The graph is concave downward.
D. Examples
1. By comparing areas, show that ln2 <1.
2. By comparing areas, show that
1
3
< ln2 <
2
4
E. The Derivative of y=ln(x)
d
1.
(ln ( x )) =
dx
x
d  1 
1
dt =
∫
dx  1 t 
x
, x >0 by the Fundamental Thm of Calculus
Therefore, for every positive value of x,
2. Applying the Chain Rule:
d
(ln ( f ( x )))
dx
d
1
(ln ( x )) =
dx
x
=
1
f ′ (x )
f ( x)
,
f ( x ) >0
F. The Integral of
1
∫ u du
1. If u is a nonzero differentiable function,
2.
e
1.
∫
1
1
∫ u du = ∫
du
= lnu + C .
u
Examples
e
dx
x
2.
∫
1
ln 2 x
dx
x
I I I . The Natural Exponential Function
A. The Inverse of ln(x) and the Function ex
1. The Function y=ex
a. ln(x) is a 1-1 increasing function, therefore, it has an inverse, ln -1x.
b. Defn : For every real number x,
ex = ln -1x.
c. ln(x) has domain (0,∞ ) and range
(−∞,∞ ) and range (0,∞ ) .
(−∞,∞ ) , ∴ ex then will have domain
d. The graph of ex is the reflection of ln(x) across the line y = x.
e.
lim e x → ∞ and
x→∞
f.
lim e x = 0 .
x →−∞
e= ln-11. e is not a rational number it is not even an algebraic number, it is a
Transcendental number like π .
B. Equations Involving ln(x) and ex
1. Inverse Equations for ex and ln(x)
a. eln x = x
; x>0
b. ln(e x) = x ;
∀x
2. Useful Operating Rules
a. To remove logarithms from an equation, exponentiate both sides.
b. To remove exponentials, take the logarithm of both sides.
C. Laws of Exponents
For all numbers x, x 1, and x2.
1. e x 1 ⋅e x2 = e x1 + x2
1
2. e − x = x
e
x1
e
3. x2 = e x1 − x2
e
4. (e x1 )
x2
= e x 1 x2 = (e x2 )
x1
D. The Derivative of ex
1. Given the function y=ex we will use logarithmic differentiation to find dy / dx.
y = ex
ln y = ln (e x )
ln y = x
1 dy
= 1
y dx
dy
= y
dx
dy
= ex
dx
2. Rules
a.
d x
e ) = ex
(
dx
b. If f (x) is any differentiable fn of x, then
(
)
d f (x )
e
= e f ( x) ⋅ f ′( x )
dx
3. Examples
Differentiate the following.
a. y = e4x+1
b.
y = lne x + e l n(6 x )
3
E. The Integral of ex
1. Since ex is its own derivative, it should be its own antiderivative:
2. Extending the rule using u-substitution:
∫ e du
u
= eu + C
∫ e dx
x
= ex + C